Property (T) for groups graded by root systems

Abstract

We introduce and study the class of groups graded by root systems. We prove that if Φ is an irreducible classical root system of rank at least 2 and G is a group graded by Φ, then under certain natural conditions on the grading, the union of the root subgroups is a Kazhdan subset of G. As the main application of this theorem we prove that for any reduced irreducible classical root system Φ of rank at least 2 and a finitely generated commutative ring R with 1, the Steinberg group St_Φ(R) and the elementary Chevalley group E_Φ(R) have property (T). We also show that there exists a group with property (T) which maps onto all finite simple groups of Lie type and rank at least 2, thereby providing a "unified" proof of expansion in these groups.

v2: 119 pages. Two new sections added (Section 9 and Appendix A); major revisions in Sections 5 and 8. In Section 9 it is proved that there exists a group with property (T) which surjects onto any finite simple group of Lie type and rank at least 2. Appendix A is based on the material from arXiv:math/0502237

Cited by in corpus (10)

• Low-dimensional representations of matrix groups and group actions on CAT(0) spaces and manifolds
• Strong algebraization of fixed point properties
• Superrigidity from Chevalley groups into acylindrically hyperbolic groups via quasi-cocycles
• On the congruence kernel of isotropic groups over rings
• Property $(TT)$ modulo $T$ and homomorphism superrigidity into mapping class groups
• On abstract representations of the groups of rational points of algebraic groups and their deformations
• Abstract homomorphisms of algebraic groups and applications
• Nonassociative algebras and groups with property ($T$)
• On Kazhdan's property (T) for isotropic reductive groups
• Property (T) for Kac-Moody groups over rings